====== Continuous Types ======

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Analysis in ''Research\Current\matingLogistics\continous20110622.nb''.
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(Revived and updated from my notebook from 2010-09-17 thru 2010-10-22.)

This topic was originally about a mathematical trick to switch continuously from one type (parameter \(p=0\)) to two (\(p=1\)).  But I found I only need to work with two types so that's what I'll focus on here.  Of course, the page title is irrelevant but that's ok.

===== Logistic processes =====

To get logistic dynamics we just need((We don't actually need \(\beta\) or \(\delta \) but it's helpful to see how they fit in.))
\[
\begin{array}{rcll}
  N   & \xrightarrow{\alpha} & 2 N       & \text{(asexual reproduction)}  \\
  2 N & \xrightarrow{\beta}  & 3 N       & \text{(sexual reproduction)}  \\
  2 N & \xrightarrow{\chi}   & N         & \text{(competition)} \\
  N   & \xrightarrow{\delta} & \emptyset & \text{(spontaneous death)}
\end{array}
\]
so
\[
\begin{array}{rcl}
  \diff{n}{t} & = & (\alpha - \delta) n - (\chi - \beta) n^2 \\
  & = & r n (1 - n/K)
\end{array}
\]
where \(r=\alpha-\delta\) and \(K=r/(\chi-\beta)\).  We require \(r,K>0\) -- equivalently, \(\alpha > \delta\) and \(\chi > \beta\) -- for a biologically meaningful outcome, with a stable, positive equilibrium population density.

Note that the above reactions capture //all// possible processes involving a single, undifferentiated type of individual and satisfying the following assumptions:
  * no higher order than pairwise interactions;
  * only one individual changes per reaction -- eg. one birth or death.

===== Extension to two types =====

Let's consider the extension of the above to two different types, \(A\) and \(B\).  We make the additional assumptions:
  * just two types, \(A\) and \(B\);
  * only the focal reactant is identified by type, any partner type is neglected; and

Then the four reactions above expand to the following 16 reactions:
\[
\begin{array}{rcll}
  A   & \xrightarrow{\alpha_{aa}} & A + A       & \text{(asexual reproduction)}  \\
  A   & \xrightarrow{\alpha_{ab}} & A + B       & \\
  B   & \xrightarrow{\alpha_{ba}} & B + A       & \\
  B   & \xrightarrow{\alpha_{bb}} & B + B       & \\
  \\

  A + N & \xrightarrow{\beta_{aa}} & A + N + A       & \text{(sexual reproduction)}  \\
  A + N & \xrightarrow{\beta_{ab}} & A + N + B       & \\
  B + N & \xrightarrow{\beta_{ba}} & B + N + A       & \\
  B + N & \xrightarrow{\beta_{bb}} & B + N + B       & \\
  \\
  
  A + N & \xrightarrow{\chi_a}   & N         & \text{(competition)} \\
  B + N & \xrightarrow{\chi_b}   & N         & \\
  \\

  A   & \xrightarrow{\delta_a} & \emptyset       & \text{(spontaneous death)}  \\
  B   & \xrightarrow{\delta_b} & \emptyset       & \\
  \\
  
  A   & \xrightarrow{\phi_a} & B             & \text{(spontaneous transition)}  \\
  B   & \xrightarrow{\phi_b} & A             & \\
  \\

  A + N & \xrightarrow{\nu_a}   & B + N         & \text{(induced transition)} \\
  B + N & \xrightarrow{\nu_b}   & A + N         &
\end{array}
\]

The first 12 reactions are extensions of the logistic processes.  The last four are new, representing transitions between the two types((These reactions may be considered implicit in the logistic processes.)).

===== Dynamics =====

Let's look at the dynamics of the two types.  From the [[wp>law of mass action]] their densities, \(a\) and \(b\), follow
\[
\begin{array}{rcl}
  \diff{a}{t} & = & \alpha_{aa} a + \alpha_{ba} b - \chi_a a n - \delta_a a + \beta_{aa} a n + \beta_{ba} b n - \phi_a a + \phi_b b - \nu_a a n + \nu_b b n \\
  \diff{b}{t} & = & \alpha_{ab} a + \alpha_{bb} b - \chi_b b n - \delta_b b + \beta_{ab} a n + \beta_{bb} b n + \phi_a a - \phi_b b + \nu_a a n - \nu_b b n
\end{array}
\]
where \(n=a+b\).

We're really interested in the dynamics of the total population density so let's switch to describing the density \(n\) and the frequency of \(a\)'s, \(f=a/n\):
\[
\begin{array}{rcl}
  \diff{n}{t} & = & \diff{a}{t} + \diff{b}{t} \\
  \diff{f}{t} & = & \frac{1}{n} \diff{a}{t} - \frac{f}{n} \diff{n}{t}.
\end{array}
\]
With substitution and some algebra we find
\[
\begin{array}{rl}
  \diff{n}{t} = & n \left[ f (\alpha_{aa} + \alpha_{ab}) + (1-f) (\alpha_{ba} + \alpha_{bb}) - f \delta_a - (1-f) \delta_b \right] \\
                & - n^2 \left[ f \chi_a + (1-f) \chi_b - f (\beta_{aa} + \beta_{ab}) - (1-f) (\beta_{ba} + \beta_{bb}) \right] \\
  \diff{f}{t} = & f (1-f) (\alpha_{aa} - \alpha_{bb}) - f^2 \alpha_{ab} + (1-f)^2 \alpha_{ba} - f (1-f) (\delta_a - \delta_b) - f \phi_a + (1-f) \phi_b \\
                & + n \left[ f (1-f) \beta_{aa} - f^2 \beta_{ab} + (1-f)^2 \beta_{ba} - f (1-f) \beta_{bb} - f (1-f) (\chi_a - \chi_b) - f \nu_a + (1-f) \nu_b \right].
\end{array}
\]

===== Separation of timescales =====

The full dynamics may be difficult to characterize.  But it is more manageable if we assume that the dynamical variables, \(n\) and \(f\), change at drastically different speeds.  For example, we can see immediately, from \(\diff{n}{t} = n(\cdot) - n^2 (\cdot)\) that if \(n\) equilibrates much faster than \(f\) -- so that \(f\) is effectively constant over the short time \(n\) equilibrates -- then the population dynamics reduce to logistic, with a slowly varying carrying capacity determined by \(f\).

On the other hand, and more formally, let's consider the case where the "ecological" processes((Those affecting the population density, \(n\).)) are very slow:
\[
\begin{array}{rlrl}
  \alpha & = \epsilon \tilde{\alpha}  &  \delta & = \epsilon \tilde{\delta} \\
  \beta  & = \epsilon \tilde{\beta}   &    \chi & = \epsilon \tilde{\chi}
\end{array}
\]
where \(\epsilon \ll 1\).

On the slow timescale, \(\tilde{t} = \epsilon t\), we have (where \(\diff{x}{t} = \epsilon x'\))
\[
\begin{array}{rl}
  n' = & n \left[ f(\tilde{\alpha}_{aa} + \tilde{\alpha}_{ab}) + (1-f) (\tilde{\alpha}_{ba} + \tilde{\alpha}_{bb}) - f \tilde{\delta}_a - (1-f) \tilde{\delta}_b \right] \\
       & - n^2 \left[ f \tilde{\chi}_a + (1-f) \tilde{\chi}_b - f (\tilde{\beta}_{aa} + \tilde{\beta}_{ab}) - (1-f) (\tilde{\beta}_{ba} + \tilde{\beta}_{bb}) \right] \\
  f' = & f (1-f) \tilde{\alpha}_{aa} - f^2 \tilde{\alpha}_{ab} + (1-f)^2 \tilde{\alpha}_{ba} - f (1-f) \tilde{\alpha}_{bb} - f(1-f)(\tilde{\delta}_a - \tilde{\delta}_b) \\
       & + n \left[ f (1-f) \tilde{\beta}_{aa} - f^2 \tilde{\beta}_{ab} + (1-f)^2 \tilde{\beta}_{ba} - f (1-f) \tilde{\beta}_{bb} - f (1-f) (\tilde{\chi}_a - \tilde{\chi}_b) \right] \\
       & + \frac{1}{\epsilon} \left[ - f \phi_a + (1-f) \phi_b - n f \nu_a + n (1-f) \nu_b \right].
\end{array}
\]

To lowest order((We can explore higher order corrections later.)) we must then have 
\[
  0 = - f \phi_a + (1-f) \phi_b - n f \nu_a + n (1-f) \nu_b 
\]
so that \(f'\) doesn't diverge.  In this case we find \(f\) quickly equilibrates (zeroth-order QSSA) to
\[
\begin{array}{rcl}
  f^*     & = & \frac{ n \nu_b + \phi_b}{n (\nu_a + \nu_b) + \phi_a + \phi_b} \\
  1 - f^* & = & \frac{ n \nu_a + \phi_a}{n (\nu_a + \nu_b) + \phi_a + \phi_b}.
\end{array}
\]
Note that we don't require //all// of the "physiological" processes (involving \(\nu\) and \(\phi\)) to be fast.  But if they are slow (on the same timescale as \(\alpha,\beta,\delta,\chi\)) then they are effectively zero in \(f^*\) for the purpose of this zeroth-order approximation.

To show deviations from logistic behaviour we need \(f^*\) to be a non-constant function of \(n\).  Otherwise, the slow dynamics reduce to \(n' = n(\cdot) - n^2(\cdot)\), the logistic equation.

The general form of the slow dynamics are
\[
  n' = \frac{n}{d_0 + d_1 n} \left( u_0 + u_1 n - u_2 n^2 \right)
\]
where((\(d_0, d_1 = 0\) are neglected because they lead to the degenerate logistic dynamics.))
\[
\begin{array}{rl}
  d_0 = & \phi_a + \phi_b > 0 \\
  d_1 = & \nu_a + \nu_b > 0 \\
  u_0 = & (\tilde{\alpha}_{aa} + \tilde{\alpha}_{ab} - \tilde{\delta}_a) \phi_b + (\tilde{\alpha}_{ba} + \tilde{\alpha}_{bb} - \tilde{\delta}_b) \phi_a \\
  u_1 = & (\tilde{\alpha}_{aa} + \tilde{\alpha}_{ab} - \tilde{\delta}_a) \nu_b + (\tilde{\alpha}_{ba} + \tilde{\alpha}_{bb} - \tilde{\delta}_b) \nu_a \\
        & + (\tilde{\beta}_{aa} + \tilde{\beta}_{ab} - \tilde{\chi}_a) \phi_b + (\tilde{\beta}_{ba} + \tilde{\beta}_{bb} - \tilde{\chi}_b) \phi_a \\
  u_2 = & (\tilde{\chi}_a - \tilde{\beta}_{aa} - \tilde{\beta}_{ab}) \nu_b + (\tilde{\chi}_b - \tilde{\beta}_{ba} - \tilde{\beta}_{bb}) \nu_a.
\end{array}
\]

===== General Allee =====

A common form of the Allee effect is
\[
  n' = r n a(n) \left( 1 - \frac{n}{K} \right)
\]
where
\[
  a(n) = 1-A + A \frac{n}{n+n_{1/2}}.
\]

Without explicitly solving the dynamics we can write our equation for \(n'\) in precisely this form by suitable choice of the parameters \(r, K, A\) and \(n_{1/2}\).  That's handy because these parameters have intuitive meaning.  \(r\) is the intrinsic growth rate (at low density, neglecting the Allee effect); \(K\) is the carrying capacity; \(A\) is the strength of the Allee effect (\(A=0\) means no effect, \(A\leq 1\) is weak, \(A>1\) is strong); \(n_{1/2}\) is the characteristic population density below which the Allee effect is significant.  See [[..:general_allee:start]] for more information.

The general Allee form can be recast as
\[
  n' = r \frac{n}{n+n_{1/2}} \left[ n_{1/2} (1-A) + n \left(1 - (1-A) \frac{n_{1/2}}{K}\right) - \frac{n^2}{K} \right].
\]

The mapping of parameters is then
\[
\begin{array}{rl}
  n_{1/2} = & \frac{d_0}{d_1} \\
  K = & \frac{u_1 \mp \sqrt{u_1^2 + 4 u_0 u_2}}{2 u_2} \\
  r = & \frac{u_1 \mp \sqrt{u_1^2 + 4 u_0 u_2}}{2 d_1} \\
  A = & \frac{2 d_0 u_2 + d_1 (u_1 \pm \sqrt{u_1^2 + 4 u_0 u_2})}{2 d_0 u_2}.
\end{array}
\]